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Self-similar processes as weak limits of a risk reserve process

100%

Burnecki K.

Probability and Mathematical Statistics

2000

tom Vol. 20, Fasc. 2

261--272

Self-similar processes are closely connected with limit theorems for identical and in general strongly dependent variables. Moreover, since they allow heavytailed distributions and provide an additional “adjusting” parameter H, they appear to be interesting in the area of risk models. In this paper we prove that only self-similar processes with stationary increments appear naturally as weak limits of a risk reserve process, and conversely every finite mean H-self-similar process with stationary increments for 0 < H ≤ 1 can result as the weak approximation. A lower bound for general self-similar processes with drift is also provided.

Recurrence theorems for Markov random walks

100%

Alsmeyer G.

Probability and Mathematical Statistics

2001

tom Vol. 21, Fasc. 1

123--134

Let (M,n, Sn)n≥0 be a Markov random walk whose driving chain (Mn)n≥0 with general state space (ℒ,Ϭ) is ergodic with unique stationary distribution ξ. Providing n−1 Sn→o in probability under Pξ, it is shown that the recurrence set of (Sn−γ(Mo) +γ(Mn))n≥o forms a closed subgroup of Rdepending on the lattice-type of (Mn, Sn)n≥o. The so-called shift function γ is bounded and appears in that lattice-type condition. The recurrence set of (Sn)n≥o itself is also given but may lookmore complicated depending on γ. The results extend the classical recurrence the orem for random walks with i.i.d. increments and further sharpenresults by Berbee, Dekking and others on the recurrence behavior of random walks with stationary increments.

Box dimension of interpolations of self-similar processes with stationary increments

75%

Herburt I.

Probability and Mathematical Statistics

2001

tom Vol. 21, Fasc. 1

171--178

We prove that under a general condition interpolation dimensions of H-sssi process converge in probability to 2−H.The result can be applied to a wideclass of H-sssi processes which includes fractional Brownian motions, (α, β)-fractional stable processes or strictly stable H-sssi processes. Moreover, we prove that for an H-sssi process with continuous sample paths the same general condition implies uniform convergence in probability of sample paths o f fractal interpolations to sample paths of the interpolated process.

75%

Maejima M. , Tudor C. A.

Probability and Mathematical Statistics

2012

tom Vol. 32, Fasc. 1

167--186

We study selfsimilar processes with stationary increments in the second Wiener chaos. We show that, in contrast with the first Wiener chaos which contains only one such process (the fractional Brownian motion), there is an infinity of selfsimilar processes with stationary increments living in the Wiener chaos of order two. We prove some limit theorems which provide a mechanism to construct such processes.